
\(\operatorname{SL}(n,k)\) ;

\(\operatorname{SL}(n/2,\)  if \(n\)  is even, where \( denotes the quaternion algebra over~\) k\(;\item \) SU(n,k,h)\(, where \) h\( is a non degenerate hermitianform on \)kn\(.\)
\(G=\operatorname{SL}(n,k)\)
We have \(|H^1(k,G(\overline{k}))|=1\)  ([1]} X §1), hence
\(L|K\)  is the unique Picard-Vessiot extension for \(\mathcal {L}(Y)=0\) , up to
\(K\) -differential isomorphism.

\(G=\operatorname{SL}(n/2,\)
We shall prove \(|H^1(k,G(\overline{k}))|=2\) , by following the same
steps as for the corollary to Proposition 3 in [1]} X §1.
Let us denote by \(1,I,J,K\)  the basis elements of \(. We recallthat \) GL(n/2,\( embeds into \) GL(n,k)\( through themorphism \) (hij) ((hij))\(, where\)
\(\mu (a+bI+cJ+dK)=\left(\begin{array}{cc} a+bi &c+di\\ -c+di & a-bi\end{array}\right), a,b,c,d \in k.\) 
We denote by \(A_n\)  the matrix \((a_{ij})_{1\le i,j \le n}\)  with
\(a_{ij}= \left\lbrace  \begin{array}{rl} 1 & \text{ \ if \ } i \text{\ is odd and \ }j=i+1, \\-1 &\text{ \ if \ }i \text{\ is even and \ } j=i-1, \\0 & \text{ \ inall other cases.}\end{array} \right.\) 
We have \(\mu (\operatorname{GL}(n/2,)=\lbrace  M \in \operatorname{GL}(n,\overline{k}) :M=A_n \overline{M} A_n^{-1} \rbrace \)  and \(\mu (\operatorname{SL}(n/2,)=\operatorname{SL}(n,\overline{k}) \cap \mu (\operatorname{GL}(n/2,)\) . We consider the action
of \(\operatorname{Gal}(\overline{k}|k)\)  on \(\operatorname{GL}(n,\overline{k})\)  given by
\(c(M)=A_n \overline{M} A_n^{-1}\) . The sequence
\(1 \rightarrow \operatorname{SL}(n,\overline{k}) \longrightarrow \operatorname{GL}(n,\overline{k})\stackrel{\det }{\longrightarrow } \overline{k}^* \rightarrow 1\) 
is an exact sequence of \(\operatorname{Gal}(\overline{k}|k)\) -modules.
With the \(\operatorname{Gal}(\overline{k}|k)\) -module structure we are
considering on \(\operatorname{GL}(n,\overline{k})\)  the proof that
\(H^1(k,\operatorname{GL}(n,\overline{k}))\)  is trivial remains valid (see
[1]} X §1) and we obtain the exact sequence
\(H^0(k,\operatorname{GL}(n,\overline{k}))\stackrel{\det }{\longrightarrow } k^*\stackrel{\delta }{\longrightarrow }H^1(k,\operatorname{SL}(n,\overline{k}))\rightarrow 1.\) 
We compute the image of
\(\det :H^0(k,\operatorname{GL}(n,\overline{k}))\rightarrow k^*\) . For \(r \in k\) ,
we have \(rId \in H^0(k,\operatorname{GL}(n,\overline{k}))=\lbrace  M \in \operatorname{GL}(n,\overline{k}) : c(M)=M \rbrace \)  and \(\det (r Id)=r^n\) , hence the
image of \(\det \)  contains all positive elements in \(k\) . On the
other hand, if \(M=c(M)\)  and \(v\)  is an eigenvector of \(M\)  with
eigenvalue \(\lambda \) , then \(A_n^{-1} \overline{v}\)  is an
eigenvector of \(M\)  with eigenvalue \(\overline{\lambda }\) , hence
\(\det M>0\) . We have then \(H^1(k,\operatorname{SL}(n,\overline{k})) \simeq \operatorname{Coker}(\det )= k^*/k^*_+\) , hence \(|H^1(k,G(\overline{k}))|=2\) . A
non trivial 1-cocycle \(x\)  of \(\operatorname{Gal}(\overline{k}|k)\)  in
\(\operatorname{SL}(n,\overline{k})\)  is given by \(x(c)= \zeta Id\) , for \(\zeta \)  a
primitive \(n\) -th root of unity. We have two Picard-Vessiot
extensions for \(\mathcal {L}(Y)=0\) , up to \(K\) -differential isomorphism. A
\(K\) -differential automorphism of \(L(i)\)  corresponding to \(x\)  is
given by the matrix \(\zeta ^{-1/2} Id\)  on the vector space of
solutions. Conjugation by \(\zeta ^{-1/2} Id\)  leaves the group
\(\operatorname{SL}(n/2,\)  stable. We obtain that the Picard-Vessiot extensions
for \(\mathcal {L}(Y)=0\)  in both \(K\) -differential isomorphy classes have
\(\operatorname{SL}(n/2,\)  as differential Galois group.

\(G=\operatorname{SU}(n,\overline{k},h)\)
It is known that if \(h\)  is a non degenerate hermitian form on
\(\overline{k}^n\) , then \(h\)  is equivalent to a hermitian form with
matrix \(I_p\) , for some integer \(p\)  with \(0\le p \le n\) , called
the index of \(h\)  and that two non degenerate hermitian forms on
\(\overline{k}^n\)  are equivalent if and only if their indexes
coincide.
We fix \(G=\lbrace  M \in \operatorname{SL}(n,\overline{k}) : \overline{M}^t I_p M =I_p \rbrace \)  and consider the action of \(\operatorname{Gal}(\overline{k}|k)\)  on
\(\operatorname{SL}(n,\overline{k})\)  given by \(c(M)=I_p (\overline{M}^t)^{-1}I_p\) . We shall prove
\(|H^1(k,G(\overline{k}))|= \left\lbrace  \begin{array}{ll} \left[ \dfrac{n}{2} \right]+1 & \text{when $n$ is odd or $p$ is even,}\\ & \\\left[ \dfrac{n}{2} \right] & \text{when $n$ is even and $p$ isodd.}\end{array} \right.\) 
A 1-cocycle from \(\operatorname{Gal}(\overline{k}|k)\)  in \(\operatorname{SL}(n,\overline{k})\) 
is given by a matrix \(B \in \operatorname{SL}(n,\overline{k})\)  satisfying \(Bc(B)= Id\) . We denote by \(x_q\)  the cocycle given by \(c \mapsto B_q\) , where \( B_q=I_q I_p\) , with \(q\)  an integer of the same
parity as \(p\)  and \(0\le q \le n\) .
We have \(x_q \sim x_{q^{\prime }} \Leftrightarrow \exists M \in \operatorname{SL}(n,\overline{k})\)  such that \(B_{q^{\prime }}=M^{-1} B_q c(M)\) . This
equality is equivalent to \(MI_{q^{\prime }}\overline{M}^t=I_q\)  which
implies \(q=q^{\prime }\) , so the 1-cocycles \(x_q\)  are pairwise non
equivalent. Let us see now that every 1-cocycle \(x\)  from
\(\operatorname{Gal}(\overline{k}|k)\)  in \(\operatorname{SL}(n,\overline{k})\)  is equivalent to
some \(x_q\) . Such a cocycle \(x\)  is determined by \(x(c)=B\) 
satisfying \(Bc(B)=Id\) , i.e.
\(BI_p=I_p\overline{B}^t=(\overline{BI_p})^t\) , so \(BI_p\)  is an
hermitian matrix, hence there exists an invertible matrix \(M\)  such
that \(M^{-1} BI_p(\overline{M}^t)^{-1}=I_r\) , for some integer \(r\) .
Equivalently
\( M^{-1} B c(M)=I_r I_p. \) 
Since \(B \in \operatorname{SL}(n,\overline{k})\) , we have \(\det (I_rI_p)=1\) , so \(x\)  is equivalent to some \(x_q\) . Let us note that,
taking determinants in (REF ), we obtain \(\det M\overline{\det M}=1\) , hence there exists \(\zeta \in \overline{k}\)  such that
\(\det (\zeta M)=1\)  and \(\zeta M\)  satisfies (REF ).
We have \(Z(\operatorname{SL}(n,\overline{k}))=\mu _n(\overline{k})\) . We want to
determine the image of \([x_q]\)  under the map
\(\varphi :H^1(k,G(\overline{k})) \rightarrow H^1(k,\operatorname{Aut}G(\overline{k}))\) . The 1-cocycle \(x_q\)  corresponds to a
Picard-Vessiot extension of \(K\)  for \(\mathcal {L}(Y)=0\)  which is determined
by an automorphism \(f_q\)  of \(L(i)\)  satisfying \(x_q=f_q^{-1}\overline{f_q}\) . We may take \(f_q\)  with matrix \(D_q:=J_qJ_p\)  over
the vector space of solutions. Since conjugation by \(D_q\)  gives the identity on \(G\) , we
obtain that all Picard-Vessiot extensions of \(K\)  for \(\mathcal {L}(Y)=0\) 
have the same differential Galois group \(G\) .

Gathering the results in this section we may state the following theorem.
Theorem 11 Let \(K\)  be a real differential field with real closed field of constants \(k\) , \(\mathcal {L}(Y)=0\)  a
linear differential equation defined over \(K\) , \(L|K\)  a real Picard-Vessiot extension for \(\mathcal {L}(Y)=0\)  and \(G\)  the differential Galois group of \(L|K\) . We assume that \(G\)  is a real form of \(\operatorname{SL}(n)\) .

If \(G=\operatorname{SL}(n,k)\) , \(L|K\)  is the unique Picard-Vessiot extension for the equation \(\mathcal {L}(Y)=0\) , up to \(K\) -differential isomorphism;

if \(G=\operatorname{SL}(n/2,\) , there are two Picard-Vessiot extensions for the equation \(\mathcal {L}(Y)=0\) , up to \(K\) -differential isomorphism, and both of them have differential Galois group \(G\) ;

if \(G=\operatorname{SU}(n,\overline{k},h)\) , there are \([n/2]+1\)  (resp. \([n/2]\) ) Picard-Vessiot extensions for the equation \(\mathcal {L}(Y)=0\) , if \(n\)  is odd or \(p\)  is even (resp. if \(n\)  is even and \(p\)  is odd), up to \(K\) -differential isomorphism, and all of them have differential Galois group \(G\) .



Forms of \(\operatorname{SO}(n), n\)  odd
This case has already been treated in [4]}. We include it
here for the sake of completeness. The real forms of \(\operatorname{SO}(n)\) ,
with \(n\)  odd, are the groups \(SO(n,k,Q)\) , where \(Q\)  is a non
degenerate quadratic form on \(k^{n}\) . The quadratic form \(Q\)  is
equivalent to a quadratic form with matrix \(I_p\) , for some integer
\(p\)  with \(0\le p \le n\) , called the index of \(Q\) , and two non
degenerate quadratic forms on \(k^n\)  are equivalent if and only if
their indexes coincide.
Write \(G=\operatorname{SO}(n,k,Q)\) , for \(Q\)  a non degenerate quadratic form on
\(k^n\)  with index \(p\) . The cohomology set \(H^1(k,G(\overline{k}))\) 
is in one-to-one correspondence with the set of equivalence
classes of quadratic forms on \(k^n\)  of rank \(n\)  and discriminant
equal to the discriminant of \(Q\) , i.e. of index of the same parity
as \(p\) . We have then \(|H^1(k,G(\overline{k}))|= \dfrac{n+1}{2}.\)
The cocycles \(x_q\)  defined by \(c \mapsto B_q\) , where \( B_q=I_qI_p\) , with \(q\)  an integer with the same parity as \(p\)  and \(0\le q\le n\) , form a complete system of representatives of the
cohomology set \(H^1(k,G(\overline{k}))\) . As in REF  we may
take the automorphism \(f_q\)  of \(L(i)\)  corresponding to \(x_q\)  to
have matrix \(D_q:=J_{2q}J_{2p}\)  over the vector space of solutions
of \(\mathcal {L}(Y)=0\) . If the matrix \(M\)  satisfies \(M^t I_p M=I_p\) , the
conjugate matrix \(N:=D_qMD_q^{-1}\)  satisfies \(N^t I_q N=I_q\) ,
hence the Picard-Vessiot extension corresponding to the 1-cocycle
\(x_q\)  has differential Galois group \(\operatorname{SO}(n,k,Q_q)\) , where \(Q_q\) 
denotes the quadratic form with index \(q\) .

The results in this section are stated in the following theorem.
Theorem 22 Let \(K\)  be a real differential field with real closed field of constants \(k\) , \(\mathcal {L}(Y)=0\)  a
linear differential equation defined over \(K\) , \(L|K\)  a real Picard-Vessiot extension for \(\mathcal {L}(Y)=0\)  and \(G\)  the differential Galois group of \(L|K\) . We assume that \(G\)  is a real form of \(\operatorname{SO}(n)\) , with \(n\)  odd. There are \((n+1)/2\)  Picard-Vessiot extensions for the equation \(\mathcal {L}(Y)=0\) , up to \(K\) -differential isomorphism, and their differential Galois groups range over the whole set of real forms of \(\operatorname{SO}(n)\) .


Forms of \(\operatorname{Sp}(2n)\)
The real forms of \(\operatorname{Sp}(2n)\)  are

\(\operatorname{Sp}(2n,k)\) ;

\(\operatorname{SU}(n,h)\) , where \(h\)  is a non degenerate hermitian form on \(n\)  (with respect to the involution \(\sigma :a+bI+cJ+dK \mapsto a-bI-cJ-dK\) ).

\(G=\operatorname{Sp}(2n,k)\)
We have \(|H^1(k,G(\overline{k}))|=1\)  ([1]} X §2, Corollary 2
), hence \(L|K\)  is the unique Picard-Vessiot extension for
\(\mathcal {L}(Y)=0\) , up to \(K\) -differential isomorphism.

\(G=\operatorname{SU}(n,h)\) , \(h\)  hermitian
If \(h\)  is a non degenerate hermitian form on \(n\) , then \(h\)  is
equivalent to a hermitian form with matrix \(I_p\)  ([6]},
7.5.3). We fix
\(G=\lbrace  M \in \operatorname{GL}(n, : \sigma (M)^tI_pM = I_p\rbrace .\) 
The group \(G\)  is the group of automorphisms of the
hermitian vector space \((n,h)\) . Let us note that for \(M,N \in M_n(\) , we have \(\sigma (MN)^t=\sigma (N)^t \sigma (M)^t\) . The set
of equivalence classes of non degenerate hermitian forms over
\(n\)  is in one-to-one correspondence with the cohomology set
\(H^1(k,G(\overline{k})\) . We have \(\operatorname{GL}(n,_k\overline{k})\simeq \operatorname{GL}(2n,\overline{k})\) , through the morphism
\(\mu \otimes _k \overline{k}\) . For \(M \in \operatorname{GL}(n,_k\overline{k})\) , we have \((\mu \otimes _k\overline{k})(\sigma (M^t))=A_n (\mu \otimes _k \overline{k})(M)^tA_n^{-1}\) . Hence \(\sigma (M)^tI_pM = I_p \Rightarrow (\mu \otimes _k\overline{k})(M)^t A_{n}^{-1} I_{2p} (\mu \otimes _k\overline{k})(M)= A_n^{-1} I_p\) . Now the group
\(\overline{G}:=\lbrace  N \in \operatorname{GL}(2n,\overline{k}) : N^t A_{n}^{-1} I_{2p} N= A_n^{-1}I_{2p} \rbrace  \) 
is a conjugate form of \(\operatorname{Sp}(2n,\overline{k})\)  and
\(\mu (G)=\lbrace  N \in \overline{G}: N=A_n \overline{N} A_n^{-1} \rbrace \) . We
have obtained that the number of \(K\) -differential isomorphic
classes of Picard-Vessiot extensions for \(\mathcal {L}(Y)=0\)  is equal to
the number of equivalence classes of non degenerate hermitian
forms over \(n\) , which is \(n+1\) . A complete set of non
equivalent 1-cocycles is given by
\( \begin{array}{lclc} x_q: & \operatorname{Gal}(\overline{k}|k)& \rightarrow & \overline{G} \\& c & \mapsto & B_q:=I_{2q} I_{2p} \end{array},\) 
with \(q\)  an integer, \(0\le q \le n\) . We may take the
automorphism \(f_q\)  of \(L(i)\)  corresponding to \(x_q\)  to have matrix
\(D_{2q}:=J_{2q}J_{2p}\)  over the vector space of solutions of \(\mathcal {L}(Y)=0\) . If
the matrix \(N\)  satisfies \(N^t A_{n}^{-1} I_{2p} N= A_n^{-1}I_{2p}\) , the conjugate matrix \(P:=D_{2q}ND_{2q}^{-1}\)  satisfies \(P^tA_{n}^{-1} I_{2q} P= A_n^{-1} I_{2q}\) , hence the Picard-Vessiot
extension corresponding to the 1-cocycle \(x_q\)  has differential
Galois group \(\operatorname{SU}(n,h_q)\) , where \(h_q\)  denotes the hermitian
form with index \(q\) .

Gathering the results in this section we may state the following theorem.
Theorem 33 Let \(K\)  be a real differential field with real closed field of constants \(k\) , \(\mathcal {L}(Y)=0\)  a
linear differential equation defined over \(K\) , \(L|K\)  a real Picard-Vessiot extension for \(\mathcal {L}(Y)=0\)  and \(G\)  the differential Galois group of \(L|K\) . We assume that \(G\)  is a real form of \(\operatorname{Sp}(2n)\) .

If \(G=\operatorname{Sp}(2n,k)\) , \(L|K\)  is the unique Picard-Vessiot extension for the equation \(\mathcal {L}(Y)=0\) ;

if \(G=\operatorname{SU}(n,h_p)\) , where \(h_p\)  is a non degenerate hermitian form on \(n\) , of index \(p\) , \(0\le p \le n\) , there are \(n+1\)  Picard-Vessiot extensions for the equation \(\mathcal {L}(Y)=0\) , up to \(K\) -differential isomorphism, and their differential Galois groups range over the whole set of groups \(G=\operatorname{SU}(n,h_q)\) , with \(h_q\)  a non degenerate hermitian form on \(n\) , of index \(q\) , \(0\le q \le n\) .



Forms of \(\operatorname{SO}(n), n\)  even
The real forms of \(\operatorname{SO}(n)\) , with \(n\)  even, are

\(\operatorname{SO}(n,k,Q)\) , where \(Q\)  is a non degenerate quadratic form on
\(k^{n}\) ;

\(\operatorname{SU}(n/2,h)\) , where \(h\)  is a non degenerate anti-hermitian form on \({n/2}\)  (with respect to the involution \(\sigma \) ).

\(G=\operatorname{SO}(n,k,Q)\)
The quadratic form \(Q\)  is equivalent to a quadratic form with
matrix \(I_p\) , for some integer \(p\)  with \(0\le p \le n\) , which
determines the equivalence class of \(Q\) .
The cohomology set \(H^1(k,G(\overline{k}))\)  is in one-to-one
correspondence with the set of equivalence classes of quadratic
forms on \(k^n\)  of rank \(n\)  and index of the same parity as \(p\) . We
have then
\(|H^1(k,G(\overline{k}))|= \left\lbrace  \begin{array}{ll} \dfrac{n}{2} +1 & \text{when $p$ is even,}\\ & \\\dfrac{n}{2} & \text{when $p$ is odd.}\end{array} \right.\) 
The cocycles \(x_q\)  defined by \(c \mapsto B_q\) , where \( B_q=I_qI_p\) , with \(q\)  an integer of the same parity as \(p\)  and \(0\le q\le n\) , form a complete system of representatives of the
cohomology set \(H^1(k,G(\overline{k}))\) . We may take the
automorphism \(f_q\)  of \(L(i)\)  corresponding to \(x_q\)  to have matrix
\(D_q:=J_qJ_p\)  over the vector space of solutions of \(\mathcal {L}(Y)=0\) . If
the matrix \(M\)  satisfies \(M^t I_p M=I_p\) , the conjugate matrix
\(N:=D_qMD_q^{-1}\)  satisfies \(N^t I_q N=I_q\) , hence the
Picard-Vessiot extension corresponding to the 1-cocycle \(x_q\)  has
differential Galois group \(\operatorname{SO}(n,k,Q_q)\) , where \(Q_q\)  denotes the
quadratic form with index \(q\) . Let us note that
\(\operatorname{SO}(n,k,Q_q)=\operatorname{SO}(n,k,Q_{n-q})\) , hence the Picard-Vessiot
extension corresponding to \(x_q\)  and \(x_{n-q}\) , \(0\le q \le (n/2)-1\) , have the same differential Galois group.

\(G=\operatorname{SU}(n/2,h)\) , \(h\)  anti-hermitian
We have \(G=\lbrace  M \in \operatorname{GL}(n/2, : \sigma (M)^t [h] M = [h]\rbrace ,\)  for
\([h]\)  the matrix of the anti-hermitian form \(h\) , in some basis of
\({n/2}\) . The group \(G\)  is the group of automorphisms of the
anti-hermitian vector space \(({n/2},h)\) . The set of equivalence
classes of non degenerate hermitian forms over \({n/2}\)  is in
one-to-one correspondence with the cohomology set
\(H^1(k,G(\overline{k}))\) . Up to equivalence, there is one single
anti-hermitian form on \(n\) , hence \(H^1(k,G(\overline{k}))=1\) . We
may check that \(\mu (G)\)  is the intersection of (a conjugate form
of) \(\operatorname{SO}(n,\overline{k})\)  with \(\mu (\operatorname{GL}(n/2,)\) . We have
obtained that \(L|K\)  is the unique Picard-Vessiot extension for
\(\mathcal {L}(Y)=0\) , up to \(K\) -differential isomorphism.

Gathering the results in this section we may state the following theorem.
Theorem 44 Let \(K\)  be a real differential field with real closed field of constants \(k\) , \(\mathcal {L}(Y)=0\)  a
linear differential equation defined over \(K\) , \(L|K\)  a real Picard-Vessiot extension for \(\mathcal {L}(Y)=0\)  and \(G\)  the differential Galois group of \(L|K\) . We assume that \(G\)  is a real form of \(\operatorname{SO}(n)\) , with \(n\)  even.

If \(G=\operatorname{SO}(n,k,Q_p)\) , where \(Q_p\)  is a non degenerate quadratic form on \(k^n\) , of index \(p\) , there are \((n/2) +1\)  (resp. \(n/2\) ) Picard-Vessiot extensions for the equation \(\mathcal {L}(Y)=0\) , up to \(K\) -differential isomorphism, when \(p\)  is even (resp. when \(p\)  is odd) and their differential Galois groups range over the whole set of groups \(G=\operatorname{SO}(n,k,Q_q)\) , with \(Q_q\)  a non degenerate quadratic form on \(k^n\) , of index \(q\) , \(0\le q \le n/2\)  and \(q\)  of the same parity as \(p\) ;

if \(G=\operatorname{SU}(n/2,h)\) , where \(h\)  is a non degenerate anti-hermitian form on \(n\) , \(L|K\)  is the unique Picard-Vessiot extension for the equation \(\mathcal {L}(Y)=0\) , up to \(K\) -differential isomorphism.



Forms of the exceptional groups
In this section we restrict our study to the groups \(G_2, F_4\)  and \(E_8\) . For \(G=G_2, F_4\)  or \(E_8\) , we have \(G \simeq \operatorname{Aut}G\) , which makes easier our study of Picard-Vessiot extensions and their Galois differential groups. For \(E_6\)  and \(E_7\)  not all real forms are explicitly described. For \(E_6\) , there are 5 real forms, of which 2 are inner. The simply connected form of \(E_6\)  can be seen as the group of transformations of an Albert algebra which leave a certain cubic form invariant. For \(E_7\) , there are 4 real forms, which are all inner. The simply connected form of \(E_7\)  can be seen as the identity component of the group of transformations of an Albert algebra leaving invariant a certain quartic form. The center of this form of \(E_7\)  has order 2 (cf. [7]} 17.7, 17,8).
\(G_2\)
The compact form of the group \(G_2\)  over a real closed field \(k\) 
is the group of automorphisms of the division algebra \(\mathbb {O}\)  of the
octonions. Hence the cohomology set \(H^1(k,G_2(\overline{k}))\)  is
in one-to-one correspondence with the set of \(k\) -isomorphism
classes of \(k\) -algebras which are \(\overline{k}\) -isomorphic to
\(\mathbb {O}\) . In turn, this set is in one-to-one correspondence with the
set of equivalence classes of 3-Pfister forms over \(k\)  (see
[7]} 17.4). A 3-Pfister form over a field \(k\)  is the
quadratic form obtained as the tensor product
\(<1,a>\otimes <1,b> \otimes <1,c>= <1,a,b,ab,c,ac,bc,abc>,\) 
with \(a,b,c \in k\setminus \lbrace 0\rbrace \) . It is easily checked
that there are two equivalence classes of 3-Pfister forms over \(k\) 
and we obtain \(|H^1(k,G_2(\overline{k}))|=2\) . Hence a linear
differential equation \(\mathcal {L}(Y)=0\)  defined over \(K\) , with
differential Galois group \(G_2\) , has two Picard-Vessiot
extensions, up to \(K\) -differential isomorphism.
We have \(\operatorname{Aut}G_2\simeq G_2\) , hence \(H^1(k,G_2(\overline{k}))\) 
classifies also the real forms of \(G_2\) . The second form (the
split form) is the group \(G_2^{\prime }\)  of automorphisms of the split
octonion algebra. Since \(\operatorname{Aut}G_2\simeq G_2\) , the map \(\Phi :H^1(k,G_2(\overline{k})) \rightarrow H^1(k,\operatorname{Aut}G_2(\overline{k}))\)  is trivially bijective and so, if \(L|K\)  is a
real Picard-Vessiot extension for a linear differential equation
\(\mathcal {L}(Y)=0\)  defined over \(K\) , with differential Galois group \(G_2\) 
(resp. \(G_2^{\prime }\) ), then the non real Picard-Vessiot extension for
\(\mathcal {L}(Y)=0\)  over \(K\)  has differential Galois group \(G_2^{\prime }\)  (resp.
\(G_2\) ).

\(F_4\)
A linear algebraic group of type \(F_4\)  is the group of
automorphisms of a so-called Albert \(k\) -algebra \(\mathbf {A}\)  (see
[7]} 17.5). We have \(|H^1(k,F_4(\overline{k}))|=3\)  and \(\operatorname{Aut}F_4 \simeq F_4\) , whence \(H^1(k,F_4(\overline{k}))\)  classifies the
real forms of \(F_4\)  up to \(k\) -isomorphism. A linear differential
equation \(\mathcal {L}(Y)=0\)  defined over \(K\) , with differential Galois
group \(F_4\) , has three Picard-Vessiot extensions, up to
\(K\) -differential isomorphism, and each of the three real forms of
\(F_4\)  appears as the differential Galois group of one of the
Picard-Vessiot extensions.

\(E_8\)
The group \(E_8\)  has three real forms, which are not easily
described. We have \(\operatorname{Aut}E_8 \simeq E_8\) , hence a linear
differential equation \(\mathcal {L}(Y)=0\)  defined over \(K\) , with
differential Galois group \(E_8\) , has three Picard-Vessiot
extensions, up to \(K\) -differential isomorphism, and each of the
three real forms of \(E_8\)  appears as the differential Galois group
of one of the Picard-Vessiot extensions.

Gathering the results in this section we may state the following theorem.
Theorem 55 Let \(K\)  be a real differential field with real closed field of constants \(k\) , \(\mathcal {L}(Y)=0\)  a
linear differential equation defined over \(K\) , \(L|K\)  a real Picard-Vessiot extension for \(\mathcal {L}(Y)=0\)  and \(G\)  the differential Galois group of \(L|K\) . We assume that \(G\)  is a real form of one of the exceptional groups \(G_2, F_4\)  or \(E_8\) .

If \(G\)  is a real form of \(G_2\) , there are 2 Picard-Vessiot extensions for the equation \(\mathcal {L}(Y)=0\) , up to \(K\) -differential isomorphism;

If \(G\)  is a real form of \(F_4\)  or \(E_8\) , there are 3 Picard-Vessiot extensions for the equation \(\mathcal {L}(Y)=0\) , up to \(K\) -differential isomorphism.

In all three cases, the differential Galois groups of the Picard-Vessiot extensions range over the whole set of real forms of \(G\times _k\overline{k}\) .



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